An offer is the initial spark of a contract; it is the seed of a contract.
An offer is said to be one of three essential elements of a contract: the other two being acceptance (of the offer) and a reciprocal flow of obligations (consideration).
Lawyers and law professors, in the legal science of contract law, will refer to the offeror and offeree as in: the offeror offers the offeree the offer.
Halsbury's Laws of England (4th Edition, 2007) defines an offer as follows:
"An offer is an expression by one person or group of persons, or by agents on his behalf, made to another, of his willingness to be bound to a contract with that other on terms certain or capable of being rendered certain."
Offers and related terms are discussed extensively in the Contract Law section.